Give 4 different pairs of polar coordinates that correspond to the Cartesian Coordinates (2,2) Aim: How do we describe Curves in Polar form?

Definition A Polar Coordinate System is a method of locating a point ( r, ) in a plane where r is a distance from a point called the pole directed at an angle θ measured counterclockwise from a polar axis.

Example Polar Grid Pole Polar Axis r=10 ( r,θ )=

r=10 r= ̶ 10 Negative radius means go in the opposite directions Remember to consider the rotation FIRST

Example ( r,θ )= Find the polar coordinates:

Example ( r,θ )= Find the polar coordinates:

Plot Name the point with three other polar coordinates using

Important Idea Polar coordinates are related to rectangular coordinates as follows: θ r y x

Example Graph: r =6 What about r = ̶ 6?

Example Graph: θ=θ= What about

Some Classic Polar Graphs Example limacon

A Classic Polar Graph Example Spiral of Archimedes

Some Classic Polar Graphs Example 8 petal rose

Use your graphing calculator to graph MODEPOL GRAPH Y= CARDIOID

Polar Roses Important Idea 8 4 petals

Polar Roses Important Idea 10 8 petals

Polar Roses Try This Name the angle in rads?

Important Idea What do you observe about the number of petals? The period?

Some Calculus Find the slope of at.

We can write polar equations parametrically where

What is this telling us about the curve?

Area r S in radians

Analysis Find the area from 0 to 0 Area 1 =

Analysis Find the area from 0 to 0 Area 2 =

Analysis Find the area from 0 to 0 Area 3 =

Analysis Find the area from 0 to 0 Riemann Sum for total area

Analysis Find the area from 0 to 0